The invention is concerned with an apparatus and a method for measuring the Theological properties of a Newtonian or a non-Newtonian fluid exhibiting a power law behavior in a tube and which is flowing in a laminar manner. This invention allows the consistency index (k) and the power law index (n) to be known in real time. It also improves the mixing of the fluid and maintains its homogeneity.
In many industrial processes, the quality of final products depends on several key physical properties, such as density, temperature, pressure, flow rate, pH, solid concentration, flow characteristics and others. These physical properties and their evolution need to be monitored and kept within given limits so as to maintain or even enhance the quality and constancy of the final products. Yet, knowing the physical properties in real time is essential in a fully automated process to provide an adequate feedback to the control system in which the target values of the parameters have been programmed. If necessary, the control system changes the amounts of ingredients or adjusts the various settings while the process is underway.
Among the physical parameters to monitor, the ones related to the flow characteristics of the materials are particularly important in a wide range of applications. Knowing the flow characteristics is the prime interest of the science of rheology. One of the key parameters in rheology is the viscosity, which may be roughly defined as the resistance to the flow of adjacent layers of a fluid in motion. All fluids exhibit viscosity to some degree. This parameter thus signals how a fluid flows under the influence of an external force or gravity. Viscosity is usually expressed in terms of Pascal-seconds or the equivalent.
Rheology characterizes fluids in two main categories, namely Newtonian fluids and non-Newtonian fluids. Sir Isaac Newton had long ago established that there is a direct linear relationship in some fluids between the shear stress (xcfx84) necessary for obtaining the movement and the effective shear rate (xcex3). The apparent viscosity (72) of these fluids is not affected by the shear rate (xcex3) and remains constant. The fluids that show this flow behavior are classified as Newtonian fluids. The ones that cannot be characterized by this kind of flow behavior are classified as non-Newtonian fluids. Some non-Newtonian fluids may have for instance a dilatant flow behavior, also referred to as a shear-thickening behavior, which is characterized by an increase in viscosity as the shear rate increases. Others may have a plastic flow behavior, also referred to as a shear-thinning behavior, characterized by a decrease in viscosity as the shear rate increases.
A power fluid is defined as any shear-thinning fluid or shear-thickening fluid having a linear relationship between log(xcex7) and (log(k)+(nxe2x88x921) log(xcex3) ), wherein xcex3 is the shear rate and xcex7 is the apparent viscosity. k and n are two rheological parameters, namely the consistency index (k) and the power law index (n). Newtonian fluids are also power law fluids since their flow behavior is a particular case of this relationship. Non-Newtonian fluids are far more complicated since they can behave as a power law fluid in one environment and not in another. For instance, it is possible to have a non-Newtonian fluid that exhibits a power law behavior when flowing in a particular pipe and not when submitted to a high shear rate, for example 104sxe2x88x921 or more.
The apparent viscosity (xcex7) of non-Newtonian fluids depends on the effective shear rate (xcex3) when the measure is taken. The value of apparent viscosity (xcex7) is thus provided with an indication of the effective shear rate (xcex3) or where the measure is taken. For instance, the apparent viscosity (xcex7) is not the same in a pipe and in a static mixer. A non-Newtonian power law fluid is more conveniently characterized by the values of the consistency index (k) and the power law index (n). The apparent viscosity (xcex7) can be calculated using the equation log(xcex7)=log(k)+(nxe2x88x921) log(xcex3). Newtonian fluids have a power law index (n) equal to 1 and the value of the consistency index (k) gives the apparent viscosity (xcex7), called the dynamic viscosity (xcexc) for these fluids.
Although apparatuses dedicated to rheological measurements in real time are found in the prior art, they cannot always be used with all fluids, particularly with fluids containing a high concentration of solid particles and which are likely to create sedimentation problems. For instance, paper coating fluid compositions generally comprise solid particles whose size is generally between 0,1 and 5.0 xcexcm. These particles are known to accumulate or migrate from walls in conventional viscometers, and thus require frequent cleaning.
In U.S. Pat. No. 4,680,957 issued on Jul. 21, 1987 and invented by Stephen C. Dodd, the consistency of a non-Newtonian fluid flowing in a laminar manner is directly calculated from a power-law model equation using a free-line pressure loss measurement. However, a shortcoming of this invention is that the pressure loss is not significant unless highly viscous and homogeneous fluids are used. If the fluid does not have a high viscosity, the value of pressure drop would not be significant enough.
The present invention provides an apparatus and a method which allows one to make accurate real time measurements of the rheological properties of a Newtonian or a non-Newtonian fluid exhibiting a power law behavior in a tube and which is flowing in a laminar manner.
More particularly, there is provided a method for determining the consistency index (k) and the power law index (n) of a fluid exhibiting a power law behavior when flowing through a pipe having an internal diameter (D), the fluid flowing in the pipe with a mean flow velocity (V), the method being characterized in that it comprises the steps of:
passing the fluid in a first and a second static mixer through which the fluid flows in a laminar manner, the first and the second static mixer being in fluid communication with each other and being non-identical, the first static mixer having predetermined geometrical constants KS1 and KP1 and the second static mixer having predetermined geometrical constants KS2 and KP2;
measuring a first pressure differential (xcex94P1) corresponding to a pressure drop of the fluid through the first static mixer;
measuring a second pressure differential (xcex94P2) corresponding to a pressure drop of the fluid through the second static mixer; and
calculating the consistency index (k) and the power law index (n) using the first and second pressure differentials (xcex94P1, xcex94P2), the mean flow velocity (V), and the geometrical constants KS1, KP1, KS2 and KP2 according to Metzner and Otto concept generalized to static mixers.
The present invention also provides an apparatus for measuring the consistency index (k) and the power law index (n) of a fluid exhibiting a power law behavior when flowing through a pipe having a given diameter (D) and with a mean flow velocity (V), the apparatus being characterized in that it comprises:
a first static mixer having an inlet and an outlet, the inlet of the first static mixer being connected to the pipe, the first static mixer having predetermined geometrical constants KS1 and KP1;
a second static mixer having an inlet, an outlet and being non-identical to the first static mixer, the outlet of the second static mixer being connected to the pipe, the second static mixer having predetermined geometrical constants KS2 and KP2;
an intermediary pipe connected between the outlet of the first static mixer and the inlet of the second static mixer;
first means for measuring a first pressure differential (xcex94P1) corresponding to a pressure drop of a laminar flow of the fluid through the first static mixer;
second means for measuring a second pressure differential (xcex94P2) corresponding to a pressure drop of the laminar flow of fluid through the second static mixer; and
third means for calculating the consistency index (k) and the power law index (n) according to Metzner and Otto concept generalized to static mixers.
The present invention also relates to an apparatus for measuring the consistency index (k) and the power law index (n) of a fluid exhibiting a power law behavior when flowing through a pipe, the apparatus being characterized in that it comprises:
a first static mixer having an inlet and an outlet, the inlet of the first static mixer being connected to the pipe, the first static mixer having predetermined geometrical constants KS1, and KP1;
a second static mixer having an inlet and an outlet, the outlet of the second static mixer being connected to the pipe, the second static mixer having predetermined geometrical constants KS2 and KP2;
an intermediary pipe mounted between the outlet of the first static mixer and the inlet of the second static mixer;
a first pressure cell mounted on the pipe before the inlet of the first static mixer;
a second pressure cell mounted on the intermediary pipe;
a third pressure cell mounted on the pipe after the outlet of the second static mixer;
a first differential pressure transducer connected between the first and the second pressure cell, the first differential pressure transducer having an output terminal for a first pressure differential signal (xcex94P1) indicative of a pressure drop of the fluid through the first static mixer;
a second differential pressure transducer connected between the second and the third pressure cell, the second differential pressure transducer having an output terminal for a second pressure differential signal (xcex94P2) indicative of a pressure drop of the fluid through the second static mixer; and
a computer having a subroutine in which the consistency index (k) and the power law index (n) are calculated according to Metzner and Otto concept generalized to static mixers.
The present invention can be used with a Newtonian fluid or a non-Newtonian power law fluid and does not involve the use of rotating parts lying in the path of the fluid and on which sedimentation may occur. If desired, determination of the rheological properties of the fluid can be made by providing the apparatus directly on the main pipe through which the fluid flows between stages of a process, for instance between the production of a paper coating fluid and the device used for applying the coating fluid as a film over a paper web. Another important advantage is that the two static mixers used in the present invention improve the homogeneity of the fluid in terms of fluid segregation, particle sedimentation, and temperature uniformity.